# How to Find the Volume of a Region Bounded by Planes and Surfaces?

• NZer
In summary, the conversation discusses finding the volume of a region bounded by four planes and a surface, with the use of integration and understanding the correct boundaries. The problem is resolved with the realization of a mistake in the diagram and the use of a simpler equation for finding the volume.
NZer
Hi guys,
Im currently revising for my exams and I encountered a problem I hope someone will be able to help me with.

## Homework Statement

Find the volume of the region of space bounded by:
The planes x=0, y=0, z=0, z=3-2x+y and the surface y=1-x^2

## Homework Equations

$$\int \int \int _R 1\, dV$$

## The Attempt at a Solution

First I decided to integrate with respect to the z direction as I wouldn't have to worry about splitting up the region yet.

$$\int \int \int _0 ^{3-2x+y} 1\, dz\, dy\, dx$$

$$= \int \int 3-2x+y\, dy \, dx$$

ok. But now I have a problem due to the surface y=1-x^2 cutting our region defined by the 4 planes. Can we split the region and choose our bounds like below?

$$= \int _1 ^{3/2} \int _{1-x^2} ^{2x-3} 3-2x+y\, dy\, dx \; + \int _0 ^1 \int _0 ^{2x-3} 3-2x+y\, dy \, dx$$

Thanks.

i have something simpler:

say f(x,y) = 3-2x+y

so now volume is

int (0,1) . int (0, 1-x^2) f(x,y) dy.dx

If you can picture the region, the volume you want appears to be confined to the 2nd octant. For this, you'll want the limits of integration for y to be from 0 to 1-x^2. Why is any "splitting of region" necessary?

Thanks for the reply rootX, Defennder

Indeed you are both right.
After scratching my head for a while I noticed I drew my diagram slightly wrong (I had the plane as z=3+2x-y lol) so my region projected onto the xy-plane was piece-wise defined.

Last edited:

## 1. What is a triple integral?

A triple integral is a mathematical concept used to calculate the volume of a three-dimensional shape. It involves integrating a function over a three-dimensional region in order to find the total volume within that region.

## 2. What are the bounds of a triple integral?

The bounds of a triple integral refer to the limits of integration for each of the three variables involved. These limits define the region over which the integration is performed and determine the total volume of the shape being evaluated.

## 3. How do I determine the bounds for a triple integral?

The bounds for a triple integral can be determined by visualizing the shape in three dimensions and identifying the limits for each variable. This can also be done using a graphing calculator or by solving a series of equations.

## 4. Can the bounds of a triple integral change?

Yes, the bounds of a triple integral can vary depending on the shape being evaluated and the specific parameters of the problem. It is important to carefully consider the bounds and make sure they accurately represent the region to be integrated.

## 5. What are some applications of triple integrals?

Triple integrals have many practical applications in fields such as physics, engineering, and economics. They are used to calculate the volume of three-dimensional objects, as well as to solve problems involving mass, density, and probability distributions.

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